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The dynamics of molecular crystal lattices. I. Neon
THE DYNAMICS
OF MOLECULAR CRYSTAL LATTICES. I. NEON. BY V. DEITZ.* INTRODUCTION.
In the application of the third law of thermodynamics to the study of chemical reactions it is important to have an extensive knowledge of the heat capacity of substances at low temperatures. Because of the experimental difficulties involved, the accumulation of this data has been rather slow, and for this reason it is highly desirable at the present time to have available methods for calculating heat capacities at low temperatures on the basis of theoretical considerations. Born and his associates have developed a very satisfactory theoretical treatment for crystals of the ionic type, but little work has been done on crystals in which the vibrating units Consist of molecules, despite the preponderance of molecular crystals in nature. In m a n y instances there is additional interest in knowing something about the molecular crystal forces because these forces play an important part in determining the chemical behavior of the compounds. This is true for organic compounds, especially for those of biochemical interest. Although the Van der Waals forces are relatively weak between a pair of atoms, which are constituent atoms of different molecules, for large molecules like the waxes and sugars, the total of the Van der Waals force over the whole molecule may be very large. For example, the energy for removing a wax molecule from its place in the crystal lattice is of the order of magnitude of several hundred kilocalories. Furthermore, forces distributed over so large an area lead to a more or less independent behavior of some particular part of the molecule * Research Fellow, The Johns Hopkins University, 1932-33; National Research Fellow, The University of Illinois, 1933-34. 459
460
V.
DE[TZ.
[J. l:. I.
which results in a decrease in those orienting influences which produce a crystalline solid. It is planned gradually to lead up to considerations of these larger molecules by first studying the simplest of molecular lattices, namely, those of the rare gases, Ne and A. T h e n in the following paper there will be discussed a diatomic example, solid N2 of the ~ modification. In a later paper the results for CH4, C2H6, and n-hexane as well as some general considerations for organic substances will be presented. LATTICE ENERGY CONSIDERATIONS.
By definition the lattice energy, ~, of a molecular crystal is simply the energy change at o ° K. for the reaction M (crystal) -4 M (gas) and m a y be given by the following equation : = 1 ( _ A + B) + E0 =
-
AHsub.,
(I)
where A is the potential energy due to the forces of attraction ; B is t h a t due to the forces of repulsion; E0 is the zero point energy; and ~Hsob. is the heat of sublimation. London 1 first pointed out t h a t the attraction potential in molecular lattices is due to a resonance potential, arising from the polarization of one atom by the electric field of the others, varying as the inverse sixth power of the distance between the atoms. Later Morgenau 2 showed t h a t an additional smaller term m u s t be added, varying inversely as the eighth power of the distance. The general molecular crystal might also have in addition the electrostatic attraction of the p e r m a n e n t dipoles present and an attraction due to an induction effect of one p e r m a n e n t dipole on the neighbors. The repulsion potential can be best represented by an exponential law of the type be -r/p where b, o are constants. T h e third term, E0, in the expression for • is the zero point energy. It constitutes an appreciable part for lattices of the lighter mass particles, in fact for neon it is about 33 per cent. of the observed heat of sublimation. 3 1 London, Zeit. Phys. Chem., BII, 222 (I93o). Morgenau, Phys. Rev., 38, 747 (I93I). 3 V. Deitz, Y. Chem. Phys., 2, 296 (I934).
Apr., I935.] D Y N A M I C S OF MOLECULAR CRYSTAL LATTICES.
46I
EVALUATION OF THE POTENTIAL CONSTANTS.
The complete lattice energy can now be given by: I[
c~° = 2
-
Y-
cij 7"06
dij
~--
7"08
+ I2be-~lP =
-
]
+ Eo
Altsub
, o r atom 0 ° K . ,
where r0 is the equilibrium distance between atoms, dii a r e constants 4 related by the expression : 3 c ij2 d,ij
-
4 e2Od
,
(:)
cii and
(3)
where e is the charge on the electron and a is the polarizability. The s u m m a t i o n is over the whole lattice. The repulsion potential decreases so rapidly with distance t h a t only the immediate twelve neighbors contribute. Slater and Kirkwood 5 have suggested the following approximation formula for the attraction constant c~-j : Cij = ( I I . 2 5 X I0-24)0~8/2n 1/'' erg-cm. 6,
where n is the n u m b e r of electrons in the external shell. The constant in this expression was determined from the more accurate calculations for hydrogen and for helium atoms. In T a b l e I a comparison is shown between the values calculated from this equation and values which have been obtained from the more precise theoretical calculations noted. The agreement is fair except in the case of the negative ions where if n were taken to be 4 (more nearly the calculated electron number) the agreement would be better, as shown in column 6. In view of the lack of good agreement the semi-empirical m e t h o d of Morgenau 6 for calculating c~j from spectroscopic data has been applied to neon and argon. To apply this m e t h o d one m u s t know the energy levels and the relative intensities of the allowed transitions from the normal state. EquXtion (6) of Morgenau's paper can be p u t into the following form for neon: 4 j. Mayer, J. Chem. Phys., x, 27o , 327 (I933). s Slater and Kirkwood, Phys. Rev., 37, 682 U930. 6 Morgenau, Phys. Rev., 37, I425 (I93t).
462
V.
[J. F. I.
DEITZ.
TABLE I. a XIO ~4.
o,673
U ...........
He . . . . . . . . . . Na ..........
•2c 5
24,2
Li + . . . . . . . . . .
0•03 ,182 ,84~ i .42 2.45
Na + ......... K + ..........
Rb + ......... C$ + .........
n.
S and K
Accurate Value
cii.
cii.
I 2 I
6.I8XI0
2
8 8 8 8
6.161M IO-6°
-60
1.49
o.I34
o.I45
0.083 2.47 •247)< io -5~ .540
0.073 i .68 •243 × IO -~s .594
1.23
1.5 2
0.273
o.145 .165 .I86 .I89 .191 .I9
(4) (4) (4) (4) (4)
.27
1.61
I.II
1.77
I.I6
1.82 1.9 ° 1.84 2.o4
1.2 5
1.33
2.04
1.33
(4) (4) (4) (4) (4) (4) (4)
1.14 1.25 1.28 1.34 1.3o 1.43 1.43
(4) (4) (4)
1.83 2.00 2.06
(4)
2'I4
(4) (4) (4)
2.I3 2'13 2'I3
0.9O .98
1.15
8 8 8 8 8 8
C l - - i n LiCI . . . . NaC1... KCI . . . . RbC1... CsC1... AgCI... TIC1 . . . .
2.95 3.09 3.30 3.29 3.22 3.45 3.45
8 8 8 8 8 8 8
B r - - i n LiBr . . . . N a B r . ,. KBr... RbBr.,. CsBr.. AgBr.. TIBr..
4.06 4.29 4.38 4.5 ° 4.47 4.47 4.47
8 8 8 8 8 8 8
2.60 2.83 2.92 3.04 3.oi 3.02 3.02
i .96 2.06 2.15 2.14 2.08 2.08
I - - i n LiI . . . . .
6.00 6.24 6.57 6.65 6.60 7.29 7•29
8 8 8 8 8 8 8
4.60 4.98 5.26 5.46 5.4 ° 6.26 6.26
3.78 3.92 4.03 4.28 4.24 4.37 4.37
1.72 3.5 ° o,39 i .63 o.87
IO 12
NaI.., KI ..... RbI.., CsI . . . . AgI . . . . TlI .... Ag + . . . . . . . . . . T1 + . . . . . . . . . . . Ne . . . . . . . . . . .
k ............ N ............
1.18 1.15
8 8 5
(6) (5) (7)
1.48
F --in LiF .... NaF... KF ..... RbF... CsF... AgF...
I.II
cii.
•309 .373 .409 .393 .39
(4)
i .30 i .29
1.85
o.8o 2.56 7.75 X IO -~° 66.2 20.4
(4) (4) (4) (4) (4) o.193 X I°-~s .218 "264 .289
.278
(4) 3.25 (4) (4)
3"52 3.72
(4) 3.85
o.67 2.68 I 1.4o5 X IO -~° 67.4 X IO ~iO
(4)
3 .8I
(4) (4)
4.41 4.41
(4)
(,4)
* F r o m this paper.
2.837 X I O - 6 0 [ ~2 ~--~/ ~--~t r~
~
~
~a~ v:~(v~+v~)
+ ~, ~',~ q~,,Ra + .o4o3"fl ] .
(4)
A p r . , I935. ]
DYNAMICS
OF
MOLECULAR
CRYSTAL
463
LATTICES.
;3 and ~, are c o n s t a n t s which m a y be d e t e r m i n e d from the two relations
dE = Zo,
fl,~ "-t~t
3 E ' 4,~ + _~ = m~? vo z
4
\ eh ]
o~,
(5)
where f i ~ -
transition probability from the normal state I to a, m = m a s s of t h e e l e c t r o n , ~1 = Ist ionization potential of neon, ~a
v~ = I -- -- where ~ = - E., the energy of state, = polarizability.
T h e distribution of the transition probabilities are assigned as follows' for the continuous region, dflE_ '
dE
the discrete levels, f ~ -
Ua3
3'
(I + E ) 3'
for
= ~¢, where J~ is the relative
intensity.
Ra
~[I =
The of the Bacher Lyman
I _[_ I 3
2Va
I in (v _t_ i) ]
Va 2
Va 3
d a t a used consider only transitions involving change main q u a n t u m number. These were t a k e n from and G o u d s m i t and from the relative intensity d a t a of and S a u n d e r s 2 TABLE II.
Transition.
Ground level .......... ISt t r a n s i t i o n . . . . . . .... 2d transition .......... 3d transition .......... 4th transition 5th transition .......... 6th transition .......... 7th transition .......... 8th transition ..........
Wave Number.
~2(2p~3s) "-~ ~1 ~4(2p53 d) ~
*l
I2,IOO
et
7,65o 6,66o 4,58o 4,426 3,37o
,6(2P55s) ~ ~1 ~7(2p56s) ~
173,930 c m . 38,760 crn. I4,77o
~8(2P~5d)~ ~
9 L y m a n a n d S a u n d e r s , Proc. Nat. Acad. Sci., I2, 93 (1926). VOL. 219, NO. I 3 1 2 - - 3 2
Relative Intensity.
IO
5 3 2.5 I. 5 I I
.3
464
V . DEITZ.
[J. V. I.
Using these d a t a the ~ ¢ ~ -- 1.811 and the s u m m a t i o n ¢--~ = 2.582" hence the equations (5) determining fl and 3'
Va2
become : 1.811/3 +
~ = 8, 2
2.582/3 + _V = 1.652 4
and
/3 = -
1.4o2
3, =
17.o8.
+
E v a l u a t i n g the s u m m a t i o n s in equation (4) we obtain the result : A2E = - 11.4o5 X IO-~° r6
Using the e q u a t i o n (3) dij = II.O X lO -76. T h e s u m m a t i o n of the a t t r a c t i o n potential over the face-centered lattice of neon is easily c o m p u t e d using the table given b y LennardJ o n e s and Ingham. 1° F r o m equation (2) and from the first derivative of potential which m u s t be zero, one obtains the following t w o relations which determine the two c o n s t a n t s p a n d b of the repulsion potential.
1 / 2 [ - 1.663 X lO -13 -t- I2be -*/~] + .098 X lO-13 = + 3.198 X 10 .5 -- I 2 - e -TIp P
.293 X lO -18
= o.
T h e value for p, given in T a b l e III, is to be c o m p a r e d with the recent theoretical value of .2o9 X lO .8 obtained b y Bleick a n d M a y e r 11 w h o investigated theoretically the repulsive potential b e t w e e n t w o like atoms. T h e i r numerical calculations for two neon a t o m s show t h a t at a distance of 3.2 A., which is the lattice c o n s t a n t of solid neon, the repulsive potential is o.4 X lO -14 ergs. Using the empirical c o n s t a n t s found a b o v e one finds t h a t the repulsive potential per pair 10Jones and Ingham, P. R. S. (London), A-II2, 214 (1926). 11Bleick and Mayer, J. Chem. Phys., 2, 252 (1934).
Apr., I935.]
DYNAMICS
OF M O L E C U L A R
CRYSTAL LATTICES.
465
of neon atoms in the crystal is 0.73 X IO-14 ergs. The greater repulsion in the crystal over that in the gas for a given distance of separation suggests that a more diffuse electron distribution function must be assigned to a neon atom in the crystal state. Presumably the high degree of degeneracy of the energy levels of the solid causes, by some exchange phenomena, this increase in the repulsive energy. The attraction potential may be expected to be only slightly dependent on the change in state because its origin is in the polarization of one atom by the electric field of the others. The behavior of this potential is classical and should be additively valid for any number of co6rdinating atoms. At any rate, this is postulated in this article. Similar considerations for argon yield the following values assembled in Table I I I. TABLE I I I .
x ro,o~. Ne . . . . [ 3.20
&HSub x Oil ,o00. x d.ij ,o,0. ~_x,o,,.' x
x a~o-. 0.39
[ 11. 4
I II.O
Io.293
A..... p379 t163 1 7.4 19,.81i , 1
E0
x po8
.o98
.276
,o,~.
_ _
.,341
[ 8.O1 X lO -1° 1 . 2 7 X IO-2
THE CRYSTAL VIBRATIONS.
Having obtained the necessary values for the constants, we investigate the dynamics of the small vibrations of a facecentered cubic lattice by the method of Born and Karman. 12 Consider a wave moving along, say, the X axis of Fig. I. Then the general equations of motion are given below, together with the corresponding solutions. The reference atom is denoted by the Miller index (l, m, n): 0 2 U l, m ,
rrt
Ot 2
n
+
X
l, m, n =
O,
0 2 V z , m,
m.
m
Ot 2
+
0 2 w z , m, n + Ot 2
Y~,m,,~
Zz,
= o,
m, ~ =
o,
U l, m, ~
~_ u e i ( v t + l $ + r a ~ + n x )
vl,,,,. ~ = Vei(vt+l~+'n¢+nx)
wt,
m, ,~ =
12 Born, " A t o m t h e o r i e der F e s t e m Zustandes."
?2)ei(*t+l~+m6+nx).
466
V. DEITZ.
[J. F. I.
u z....... v z. . . . . . and w~. . . . . are the displacements of the particle at (1, m, n) in the X, Y, Z direction respectively; X z. . . . . Yz, m, n, Z~ . . . . is the force set up in the X, Y, Z direction respectively; finally 6, ~, x are the "phase-comp o n e n t s " of the wave (see Born, 1. c.). W e now denote by a, the force c o n s t a n t between the reference a t o m and the neighboring a t o m s in the X Z plane; FIG. I.
by B the c o n s t a n t between the reference a t o m and its neighbors in YZ; b y 3' the c o n s t a n t between the reference a t o m and its neighbors in X Y; b y 6 the c o n s t a n t between the reference a t o m and the next closest neighbor in the X Z plane ; finally b y t h a t between the reference a t o m and the four next closest neighbors in the YZ plane. T h e n the following expression can be derived for X~, m, ~ ; expressions for Yz . . . . a n d Zl . . . . m a y be w r i t t e n b y cyclic p e r m u t a t i o n of the indices:
Apr., 1935.]
Xl ....
=
DYNAMICS
OF ~ I O L E C U L A R
OL(Ul+I . . . . .
+1 +
Ul--1 . . . . .
CRYSTAL +1 + +
Ul+I . . . . . .
u~-I .....
467
LATTICES.
1
1 -- 4uz,
m, n)
-'1- / ~ ( U l , re+l, n + l -1- U l , m--l, n + l J[- Ul, re+l, n--1
"JI- Ul, m--l, n--1 - - 4 U l . . . . -J[- V(Ul-t-1, re+l, n "JI- Ul--1, re+l, n +
)
U / + I , m--l, n
"'q-- I/'Z--1, m--l, n - -
4ut . . . . )
+ ~(u~+~ . . . . + ut_~ ..... - 2Uz . . . . . . ) + ~ ( u t , ..+~, ,, + u t . . . . . 2, . + u l . . . . . +~ Jr U z.....
2 -- 41tt .....
).
S u b s t i t u t i n g the solution into the equation we obtain the secular equation, which cab be simplified by the trigonometric relation : e ix - ~ e - i x =
COS X .
2
Also, from the definition of ¢, 4, x a n d from the fact t h a t we are considering a wave moving along the X axis: 2 ~-a
¢ =-~-p,
~=o,
x=o
wherein (p, q, r) are the cosines of the wave, a n d p = I, q = r = o; X is the wave-length a n d a is the lattice constant. T h e following two equations for the frequencies are obtained : -- mvz2 = (a + 3")
( (4cos 4 ) --
--my, ~ = (3'+/3)
4cos--~---
4
+ ~
2cos x -
2
-]-~
2cos-~---
2
) •
The ¢ can be w r i t t e n as follows: ~/~ 71"~ -- = -=
2
X
71"K
2 ( N + i)
T h e individual wave-lengths are o b t a i n e d by allowing K to t a k e on values up to N + I, where N is the n u m b e r of particles on an edge of the crystal. Hence the fastest vibration occurs when K ~ N + I or ¢ ~ ~r and we have for the frequency in sec. -1 instead of (2rr) sec. -1 the following:
468
V.
DEITZ.
[J. F. I.
x P m a x (l)
71"
Plnax (t)
71"
Here the frequency denoted by v z is identified as the longitudinal frequency because it entails the proper force constants, v t is the transverse vibration. It m a y be seen that the fastest mode of vibration is such that the atoms in some one plane move in the same direction with its immediate neighbors completely out of phase; this also is obtained from the force constants involved. To evaluate the force constant a, which obviously is equal to ~', we need only consider a single pair of atoms, since the force constants were so defined. Referring to Fig. 2, the FIG. 2.
4.0 J J
X ,,.,s %
4-o
force constant for the mode of motion considered is given as follows : a=Ofir2
~
+\roar!
I -~
•
Apr., 1935.]
DYNAMICS
OF M O L E C U L A R
CRYSTAL
LATTICES.
469
Here I 06#
6Cij + =
024
+
;[
Or i =
ro 1°
,
ro 8
42c~. ro 8
rop 72d~j r0 TM + -
b2e ro/o] -
,
Since x2/ro ~ = ½ we obtain by evaluating the first and second derivatives the results in Table IV. TABLE I V . ro Or I
Ne
06"
-- 0.007 X I0 3 -- O.OII X IO 3
+ o.234 )< IO3 + 1.425 >( IO3
o . I I 4 X IO3 o.7o7 X IO3
98
63 85
The maximum longitudinal frequency which has been calculated is to be identified with the Debye 0..... Any other frequency which might be calculated would be less than this. The agreement between the theoretical 0m~ and the value calculated from heat capacity data is all that one can expect when the data used is viewed in retrospect. The author takes this opportunity to thank Professor D. H. Andrews for advice and suggestions. URBANA~ILLINOIS, S e p t e m b e r 20, I934.
OF MOLECULAR CRYSTAL LATTICES. I. NEON. BY V. DEITZ.* INTRODUCTION.
In the application of the third law of thermodynamics to the study of chemical reactions it is important to have an extensive knowledge of the heat capacity of substances at low temperatures. Because of the experimental difficulties involved, the accumulation of this data has been rather slow, and for this reason it is highly desirable at the present time to have available methods for calculating heat capacities at low temperatures on the basis of theoretical considerations. Born and his associates have developed a very satisfactory theoretical treatment for crystals of the ionic type, but little work has been done on crystals in which the vibrating units Consist of molecules, despite the preponderance of molecular crystals in nature. In m a n y instances there is additional interest in knowing something about the molecular crystal forces because these forces play an important part in determining the chemical behavior of the compounds. This is true for organic compounds, especially for those of biochemical interest. Although the Van der Waals forces are relatively weak between a pair of atoms, which are constituent atoms of different molecules, for large molecules like the waxes and sugars, the total of the Van der Waals force over the whole molecule may be very large. For example, the energy for removing a wax molecule from its place in the crystal lattice is of the order of magnitude of several hundred kilocalories. Furthermore, forces distributed over so large an area lead to a more or less independent behavior of some particular part of the molecule * Research Fellow, The Johns Hopkins University, 1932-33; National Research Fellow, The University of Illinois, 1933-34. 459
460
V.
DE[TZ.
[J. l:. I.
which results in a decrease in those orienting influences which produce a crystalline solid. It is planned gradually to lead up to considerations of these larger molecules by first studying the simplest of molecular lattices, namely, those of the rare gases, Ne and A. T h e n in the following paper there will be discussed a diatomic example, solid N2 of the ~ modification. In a later paper the results for CH4, C2H6, and n-hexane as well as some general considerations for organic substances will be presented. LATTICE ENERGY CONSIDERATIONS.
By definition the lattice energy, ~, of a molecular crystal is simply the energy change at o ° K. for the reaction M (crystal) -4 M (gas) and m a y be given by the following equation : = 1 ( _ A + B) + E0 =
-
AHsub.,
(I)
where A is the potential energy due to the forces of attraction ; B is t h a t due to the forces of repulsion; E0 is the zero point energy; and ~Hsob. is the heat of sublimation. London 1 first pointed out t h a t the attraction potential in molecular lattices is due to a resonance potential, arising from the polarization of one atom by the electric field of the others, varying as the inverse sixth power of the distance between the atoms. Later Morgenau 2 showed t h a t an additional smaller term m u s t be added, varying inversely as the eighth power of the distance. The general molecular crystal might also have in addition the electrostatic attraction of the p e r m a n e n t dipoles present and an attraction due to an induction effect of one p e r m a n e n t dipole on the neighbors. The repulsion potential can be best represented by an exponential law of the type be -r/p where b, o are constants. T h e third term, E0, in the expression for • is the zero point energy. It constitutes an appreciable part for lattices of the lighter mass particles, in fact for neon it is about 33 per cent. of the observed heat of sublimation. 3 1 London, Zeit. Phys. Chem., BII, 222 (I93o). Morgenau, Phys. Rev., 38, 747 (I93I). 3 V. Deitz, Y. Chem. Phys., 2, 296 (I934).
Apr., I935.] D Y N A M I C S OF MOLECULAR CRYSTAL LATTICES.
46I
EVALUATION OF THE POTENTIAL CONSTANTS.
The complete lattice energy can now be given by: I[
c~° = 2
-
Y-
cij 7"06
dij
~--
7"08
+ I2be-~lP =
-
]
+ Eo
Altsub
, o r atom 0 ° K . ,
where r0 is the equilibrium distance between atoms, dii a r e constants 4 related by the expression : 3 c ij2 d,ij
-
4 e2Od
,
(:)
cii and
(3)
where e is the charge on the electron and a is the polarizability. The s u m m a t i o n is over the whole lattice. The repulsion potential decreases so rapidly with distance t h a t only the immediate twelve neighbors contribute. Slater and Kirkwood 5 have suggested the following approximation formula for the attraction constant c~-j : Cij = ( I I . 2 5 X I0-24)0~8/2n 1/'' erg-cm. 6,
where n is the n u m b e r of electrons in the external shell. The constant in this expression was determined from the more accurate calculations for hydrogen and for helium atoms. In T a b l e I a comparison is shown between the values calculated from this equation and values which have been obtained from the more precise theoretical calculations noted. The agreement is fair except in the case of the negative ions where if n were taken to be 4 (more nearly the calculated electron number) the agreement would be better, as shown in column 6. In view of the lack of good agreement the semi-empirical m e t h o d of Morgenau 6 for calculating c~j from spectroscopic data has been applied to neon and argon. To apply this m e t h o d one m u s t know the energy levels and the relative intensities of the allowed transitions from the normal state. EquXtion (6) of Morgenau's paper can be p u t into the following form for neon: 4 j. Mayer, J. Chem. Phys., x, 27o , 327 (I933). s Slater and Kirkwood, Phys. Rev., 37, 682 U930. 6 Morgenau, Phys. Rev., 37, I425 (I93t).
462
V.
[J. F. I.
DEITZ.
TABLE I. a XIO ~4.
o,673
U ...........
He . . . . . . . . . . Na ..........
•2c 5
24,2
Li + . . . . . . . . . .
0•03 ,182 ,84~ i .42 2.45
Na + ......... K + ..........
Rb + ......... C$ + .........
n.
S and K
Accurate Value
cii.
cii.
I 2 I
6.I8XI0
2
8 8 8 8
6.161M IO-6°
-60
1.49
o.I34
o.I45
0.083 2.47 •247)< io -5~ .540
0.073 i .68 •243 × IO -~s .594
1.23
1.5 2
0.273
o.145 .165 .I86 .I89 .191 .I9
(4) (4) (4) (4) (4)
.27
1.61
I.II
1.77
I.I6
1.82 1.9 ° 1.84 2.o4
1.2 5
1.33
2.04
1.33
(4) (4) (4) (4) (4) (4) (4)
1.14 1.25 1.28 1.34 1.3o 1.43 1.43
(4) (4) (4)
1.83 2.00 2.06
(4)
2'I4
(4) (4) (4)
2.I3 2'13 2'I3
0.9O .98
1.15
8 8 8 8 8 8
C l - - i n LiCI . . . . NaC1... KCI . . . . RbC1... CsC1... AgCI... TIC1 . . . .
2.95 3.09 3.30 3.29 3.22 3.45 3.45
8 8 8 8 8 8 8
B r - - i n LiBr . . . . N a B r . ,. KBr... RbBr.,. CsBr.. AgBr.. TIBr..
4.06 4.29 4.38 4.5 ° 4.47 4.47 4.47
8 8 8 8 8 8 8
2.60 2.83 2.92 3.04 3.oi 3.02 3.02
i .96 2.06 2.15 2.14 2.08 2.08
I - - i n LiI . . . . .
6.00 6.24 6.57 6.65 6.60 7.29 7•29
8 8 8 8 8 8 8
4.60 4.98 5.26 5.46 5.4 ° 6.26 6.26
3.78 3.92 4.03 4.28 4.24 4.37 4.37
1.72 3.5 ° o,39 i .63 o.87
IO 12
NaI.., KI ..... RbI.., CsI . . . . AgI . . . . TlI .... Ag + . . . . . . . . . . T1 + . . . . . . . . . . . Ne . . . . . . . . . . .
k ............ N ............
1.18 1.15
8 8 5
(6) (5) (7)
1.48
F --in LiF .... NaF... KF ..... RbF... CsF... AgF...
I.II
cii.
•309 .373 .409 .393 .39
(4)
i .30 i .29
1.85
o.8o 2.56 7.75 X IO -~° 66.2 20.4
(4) (4) (4) (4) (4) o.193 X I°-~s .218 "264 .289
.278
(4) 3.25 (4) (4)
3"52 3.72
(4) 3.85
o.67 2.68 I 1.4o5 X IO -~° 67.4 X IO ~iO
(4)
3 .8I
(4) (4)
4.41 4.41
(4)
(,4)
* F r o m this paper.
2.837 X I O - 6 0 [ ~2 ~--~/ ~--~t r~
~
~
~a~ v:~(v~+v~)
+ ~, ~',~ q~,,Ra + .o4o3"fl ] .
(4)
A p r . , I935. ]
DYNAMICS
OF
MOLECULAR
CRYSTAL
463
LATTICES.
;3 and ~, are c o n s t a n t s which m a y be d e t e r m i n e d from the two relations
dE = Zo,
fl,~ "-t~t
3 E ' 4,~ + _~ = m~? vo z
4
\ eh ]
o~,
(5)
where f i ~ -
transition probability from the normal state I to a, m = m a s s of t h e e l e c t r o n , ~1 = Ist ionization potential of neon, ~a
v~ = I -- -- where ~ = - E., the energy of state, = polarizability.
T h e distribution of the transition probabilities are assigned as follows' for the continuous region, dflE_ '
dE
the discrete levels, f ~ -
Ua3
3'
(I + E ) 3'
for
= ~¢, where J~ is the relative
intensity.
Ra
~[I =
The of the Bacher Lyman
I _[_ I 3
2Va
I in (v _t_ i) ]
Va 2
Va 3
d a t a used consider only transitions involving change main q u a n t u m number. These were t a k e n from and G o u d s m i t and from the relative intensity d a t a of and S a u n d e r s 2 TABLE II.
Transition.
Ground level .......... ISt t r a n s i t i o n . . . . . . .... 2d transition .......... 3d transition .......... 4th transition 5th transition .......... 6th transition .......... 7th transition .......... 8th transition ..........
Wave Number.
~2(2p~3s) "-~ ~1 ~4(2p53 d) ~
*l
I2,IOO
et
7,65o 6,66o 4,58o 4,426 3,37o
,6(2P55s) ~ ~1 ~7(2p56s) ~
173,930 c m . 38,760 crn. I4,77o
~8(2P~5d)~ ~
9 L y m a n a n d S a u n d e r s , Proc. Nat. Acad. Sci., I2, 93 (1926). VOL. 219, NO. I 3 1 2 - - 3 2
Relative Intensity.
IO
5 3 2.5 I. 5 I I
.3
464
V . DEITZ.
[J. V. I.
Using these d a t a the ~ ¢ ~ -- 1.811 and the s u m m a t i o n ¢--~ = 2.582" hence the equations (5) determining fl and 3'
Va2
become : 1.811/3 +
~ = 8, 2
2.582/3 + _V = 1.652 4
and
/3 = -
1.4o2
3, =
17.o8.
+
E v a l u a t i n g the s u m m a t i o n s in equation (4) we obtain the result : A2E = - 11.4o5 X IO-~° r6
Using the e q u a t i o n (3) dij = II.O X lO -76. T h e s u m m a t i o n of the a t t r a c t i o n potential over the face-centered lattice of neon is easily c o m p u t e d using the table given b y LennardJ o n e s and Ingham. 1° F r o m equation (2) and from the first derivative of potential which m u s t be zero, one obtains the following t w o relations which determine the two c o n s t a n t s p a n d b of the repulsion potential.
1 / 2 [ - 1.663 X lO -13 -t- I2be -*/~] + .098 X lO-13 = + 3.198 X 10 .5 -- I 2 - e -TIp P
.293 X lO -18
= o.
T h e value for p, given in T a b l e III, is to be c o m p a r e d with the recent theoretical value of .2o9 X lO .8 obtained b y Bleick a n d M a y e r 11 w h o investigated theoretically the repulsive potential b e t w e e n t w o like atoms. T h e i r numerical calculations for two neon a t o m s show t h a t at a distance of 3.2 A., which is the lattice c o n s t a n t of solid neon, the repulsive potential is o.4 X lO -14 ergs. Using the empirical c o n s t a n t s found a b o v e one finds t h a t the repulsive potential per pair 10Jones and Ingham, P. R. S. (London), A-II2, 214 (1926). 11Bleick and Mayer, J. Chem. Phys., 2, 252 (1934).
Apr., I935.]
DYNAMICS
OF M O L E C U L A R
CRYSTAL LATTICES.
465
of neon atoms in the crystal is 0.73 X IO-14 ergs. The greater repulsion in the crystal over that in the gas for a given distance of separation suggests that a more diffuse electron distribution function must be assigned to a neon atom in the crystal state. Presumably the high degree of degeneracy of the energy levels of the solid causes, by some exchange phenomena, this increase in the repulsive energy. The attraction potential may be expected to be only slightly dependent on the change in state because its origin is in the polarization of one atom by the electric field of the others. The behavior of this potential is classical and should be additively valid for any number of co6rdinating atoms. At any rate, this is postulated in this article. Similar considerations for argon yield the following values assembled in Table I I I. TABLE I I I .
x ro,o~. Ne . . . . [ 3.20
&HSub x Oil ,o00. x d.ij ,o,0. ~_x,o,,.' x
x a~o-. 0.39
[ 11. 4
I II.O
Io.293
A..... p379 t163 1 7.4 19,.81i , 1
E0
x po8
.o98
.276
,o,~.
_ _
.,341
[ 8.O1 X lO -1° 1 . 2 7 X IO-2
THE CRYSTAL VIBRATIONS.
Having obtained the necessary values for the constants, we investigate the dynamics of the small vibrations of a facecentered cubic lattice by the method of Born and Karman. 12 Consider a wave moving along, say, the X axis of Fig. I. Then the general equations of motion are given below, together with the corresponding solutions. The reference atom is denoted by the Miller index (l, m, n): 0 2 U l, m ,
rrt
Ot 2
n
+
X
l, m, n =
O,
0 2 V z , m,
m.
m
Ot 2
+
0 2 w z , m, n + Ot 2
Y~,m,,~
Zz,
= o,
m, ~ =
o,
U l, m, ~
~_ u e i ( v t + l $ + r a ~ + n x )
vl,,,,. ~ = Vei(vt+l~+'n¢+nx)
wt,
m, ,~ =
12 Born, " A t o m t h e o r i e der F e s t e m Zustandes."
?2)ei(*t+l~+m6+nx).
466
V. DEITZ.
[J. F. I.
u z....... v z. . . . . . and w~. . . . . are the displacements of the particle at (1, m, n) in the X, Y, Z direction respectively; X z. . . . . Yz, m, n, Z~ . . . . is the force set up in the X, Y, Z direction respectively; finally 6, ~, x are the "phase-comp o n e n t s " of the wave (see Born, 1. c.). W e now denote by a, the force c o n s t a n t between the reference a t o m and the neighboring a t o m s in the X Z plane; FIG. I.
by B the c o n s t a n t between the reference a t o m and its neighbors in YZ; b y 3' the c o n s t a n t between the reference a t o m and its neighbors in X Y; b y 6 the c o n s t a n t between the reference a t o m and the next closest neighbor in the X Z plane ; finally b y t h a t between the reference a t o m and the four next closest neighbors in the YZ plane. T h e n the following expression can be derived for X~, m, ~ ; expressions for Yz . . . . a n d Zl . . . . m a y be w r i t t e n b y cyclic p e r m u t a t i o n of the indices:
Apr., 1935.]
Xl ....
=
DYNAMICS
OF ~ I O L E C U L A R
OL(Ul+I . . . . .
+1 +
Ul--1 . . . . .
CRYSTAL +1 + +
Ul+I . . . . . .
u~-I .....
467
LATTICES.
1
1 -- 4uz,
m, n)
-'1- / ~ ( U l , re+l, n + l -1- U l , m--l, n + l J[- Ul, re+l, n--1
"JI- Ul, m--l, n--1 - - 4 U l . . . . -J[- V(Ul-t-1, re+l, n "JI- Ul--1, re+l, n +
)
U / + I , m--l, n
"'q-- I/'Z--1, m--l, n - -
4ut . . . . )
+ ~(u~+~ . . . . + ut_~ ..... - 2Uz . . . . . . ) + ~ ( u t , ..+~, ,, + u t . . . . . 2, . + u l . . . . . +~ Jr U z.....
2 -- 41tt .....
).
S u b s t i t u t i n g the solution into the equation we obtain the secular equation, which cab be simplified by the trigonometric relation : e ix - ~ e - i x =
COS X .
2
Also, from the definition of ¢, 4, x a n d from the fact t h a t we are considering a wave moving along the X axis: 2 ~-a
¢ =-~-p,
~=o,
x=o
wherein (p, q, r) are the cosines of the wave, a n d p = I, q = r = o; X is the wave-length a n d a is the lattice constant. T h e following two equations for the frequencies are obtained : -- mvz2 = (a + 3")
( (4cos 4 ) --
--my, ~ = (3'+/3)
4cos--~---
4
+ ~
2cos x -
2
-]-~
2cos-~---
2
) •
The ¢ can be w r i t t e n as follows: ~/~ 71"~ -- = -=
2
X
71"K
2 ( N + i)
T h e individual wave-lengths are o b t a i n e d by allowing K to t a k e on values up to N + I, where N is the n u m b e r of particles on an edge of the crystal. Hence the fastest vibration occurs when K ~ N + I or ¢ ~ ~r and we have for the frequency in sec. -1 instead of (2rr) sec. -1 the following:
468
V.
DEITZ.
[J. F. I.
x P m a x (l)
71"
Plnax (t)
71"
Here the frequency denoted by v z is identified as the longitudinal frequency because it entails the proper force constants, v t is the transverse vibration. It m a y be seen that the fastest mode of vibration is such that the atoms in some one plane move in the same direction with its immediate neighbors completely out of phase; this also is obtained from the force constants involved. To evaluate the force constant a, which obviously is equal to ~', we need only consider a single pair of atoms, since the force constants were so defined. Referring to Fig. 2, the FIG. 2.
4.0 J J
X ,,.,s %
4-o
force constant for the mode of motion considered is given as follows : a=Ofir2
~
+\roar!
I -~
•
Apr., 1935.]
DYNAMICS
OF M O L E C U L A R
CRYSTAL
LATTICES.
469
Here I 06#
6Cij + =
024
+
;[
Or i =
ro 1°
,
ro 8
42c~. ro 8
rop 72d~j r0 TM + -
b2e ro/o] -
,
Since x2/ro ~ = ½ we obtain by evaluating the first and second derivatives the results in Table IV. TABLE I V . ro Or I
Ne
06"
-- 0.007 X I0 3 -- O.OII X IO 3
+ o.234 )< IO3 + 1.425 >( IO3
o . I I 4 X IO3 o.7o7 X IO3
98
63 85
The maximum longitudinal frequency which has been calculated is to be identified with the Debye 0..... Any other frequency which might be calculated would be less than this. The agreement between the theoretical 0m~ and the value calculated from heat capacity data is all that one can expect when the data used is viewed in retrospect. The author takes this opportunity to thank Professor D. H. Andrews for advice and suggestions. URBANA~ILLINOIS, S e p t e m b e r 20, I934.